Top Sets and Top Cycles:

A fundamental property of top sets is that a top set of a tight
immersion is itself a tight immersion
[K4]. A surprising result is
that a top set need not be an embedding, as shown below.

This is the topset for a three-handled
torus:
the torus itself is formed by taking the boundary of the figure
(black line) cross an interval and pasting a copy of the top
set at each end of the tube so formed. The outer ring forms a torus,
and the two cross pieces form two additional handles.

In general, the interesting top sets are those containing the top
cycles. For smooth surfaces, distinct top sets must be disjoint, but
for polyhedral surfaces, different top sets can share points or even
edges, so the top cycles need not be distinct.

To see that the last is true, note that, in the smooth case, since the
remainder of the surface lies on one side of the support plane for the
top cycle, the surface is tangent to the plane, so for each point of
the top cycle, the surface is locally the graph of a function with
respect to the plane; thus in a small enough neighborhood of the top
cycle, the projection of the surface onto the plane is an embedding.
Since a
Möbius band
can not be embedded in the plane, the neighborhood of the top cycle
must be orientable.

Another way to see this is to consider the supporting plane for the
top cycle, and imagine pushing the plane down (through the surface).
As it moves, it intersects the surface in two curves, one on each side
of the top cycle. These curves trace out a neighborhood that is
an orientable strip (since it has two boundary curves). This holds
for both smooth and polyhedral surfaces.

The Number of Top Cycles:

Cecil and Ryan
[CR1] showed that the number
of top cycles of a tight immersion (of a
connected
surface other than
a sphere) is always at least 2 and no more than 2 - X(M),
where X(M) is the
Euler characteristic
of the surface.

The lower bound is easy to see (using the decomposition of the surface
into M+ and M- regions
):
if a tight immersion of a connected surface has no top cycles, then it
is necessarily a sphere, since in this case the image of the M+
region is all of the convex envelope (and there is no M-
region). So a tight immersion of any surface other than the sphere
must have at least one top cycle. On the other hand, it can't have
just one, since such a top cycle would bound a region (namely, the
region M+), so there must be at least two top cycles.

The upper bound comes from the fact that the Euler characteristic is
tied to the
genus
of the surface, and the genus limits the number of tubes that can be
used to attach the interior M- region to the outer M+
region along the top cycles.

Tightness and polar height functionsTightness and the convex hullTightness and its consequences